Superderivations ofC *-algebras implemented by symmetric operators

The paper studies unbounded symmetric and dissipative implementations (S,G) of^*-superderivations ofC ^*-algebras . It associates with them representations _S of the domainsD() of on the deficiency spacesN(S) of the symmetric operatorsS. A link is obtained between the deficiency indicesn _±(S) ofS and the dimensions of irreducible representations of . For the case when (S,G) is a maximal implementation and max(n _±(S))<, some conditions are given for the representation _S to be semisimple and to extend to a bounded representation of .     

Markov symmetric stable processes

Using the method of generative processes, we construct a model of a random linear symmetric stable Markov process, which is a natural generalization of the Markov Gaussian process and preserves its main property: invariance with respect to arbitrary linear transformations. Methods for analyzing such processes are developed. In particular, it is proposed to use the information correlation function as a characteristic of the pair dependence between the process values at different times. This function is used to calculate the determination interval of the process. Voronezh State University, Voronezh, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 3, pp. 264–270, March, 2000     

Dirichlet forms and Markov semigroups onC*-algebras

We extend the classical theory of Dirichlet forms and associated Markov semigroups to the case of aC*-algebra with a trace. Semigroups of completely positive maps are characterized by completely positive Dirichlet forms.Work supported by The Norwegian Research Council for Science and the Humanities     

Markov quantum semigroups admit covariant MarkovC*-dilations

Through a Daniell-Kolmogorov type construction, to any Markov quantum semigroup on aC*-algebra there is associated a quantum stochastic process which is a dilation of the semigroup, and satisfies a covariance rule which implies the weak Markov property.     

Endomorphism semigroups and lightlike translations

Borchers and Wiesbrock have studied the one-parameter semigroups of endomorphisms of von Neumann algebras that appear as lightlike translations in the theory of algebras of local observables, showing that they automatically transform under the appropriate modular automorphisms as under velocity transformations. Here, these results are abstracted and analyzed as essentially operator-theoretic. Criteria are then established for a spatial derivation of a von Neumann algebra to generate a one-parameter semigroup of endomorphisms, and all of this is combined to establish a von Neumann-algebraic converse to the Borchers and Wiesbrock results. This sort of analysis is then applied to questions of isotony and covariance for local algebras, to show that Poincaré covariance together with a domain condition for the translations can imply isotony.This research was partly supported by a fellowship from the Consiglio Nazionale delle Ricerche.     

Partial Abelian semigroups

Orthomodular lattices and posets, orthoalgebras, and D-posets are all examples of partial Abelian semigroups. So, too, are the event structures of test spaces. The passage from an algebraic test space to its logic (an orthoalgebra) is an instance of a general construction involving a partial Abelian semigroupL and a distinguished subsetM L such that perspectivity with respect toM is a congruence onL. The quotient ofL by such a congruence is always a cancellative, unital PAS, and every such PAS arises canonically as such a quotient.     

Binary quantum logic and generating semigroups

A refined definition of basic concepts for logic describing physical systems is proposed. Within the suggested formalism of generating semigroups the active logic of questions and passive logic of answers are introduced. The objects for which both logics are isomorphic are called self-adequate. It is shown that the assumption of self-adequacy implies that the object is either quantum or classical. The possibility of application of the theory to non-self-adequate objects is discussed.     

Asymptotically normal dynamical semigroups

A definition and a characterization of asymptotically normal dynamical systems are given. In particular, a theorem concerning the return to equilibrium is presented.     

Irreducible quantum dynamical semigroups

We study some irreducible and ergodic properties of quantum dynamical semigroups, and apply our methods to semigroups of Lindblad type.     

Remarks on positive semigroups

Let ℳ be a von Neumann algebra with a cyclic and separating vector Ω and let ω(·) denote the corresponding vector state, i.e., ω(A)=(Ω, AΩ) A ∈ ℳ. We have proved that a positive semigroup τ on ℳ can induce the dynamical semigroup in the GNS representation associated with ω if the state ω is a τ-invariant one. Some applications are given.     

Perturbations of Gibbs semigroups

We present the analytic perturbation theory for Gibbs semigroups in the case when perturbations of generators are relatively bounded. Analyticity with respect to perturbation and semigroup parameters in the Tr-norm topology is proved and the corresponding domains are described.     

Quantum dynamical semigroups and approach to equilibrium

For a quantum dynamical semigroup possessing a faithful normal stationary state, some conditions are discussed, which ensure the uniqueness of the equilibrium state and/or the approach to equilibrium for arbitrary initial condition.A fellowship from the Italian Ministry of Public Education is acknowledged.     

Strongly positive semigroups and faithful invariant states

Let (?, τ, ω) denote aW*-algebra ?, a semigroupt>0?τ _t of linear maps of ? into ?, and a faithful τ-invariant normal state ω over ?. We assume that τ is strongly positive in the sense that $$\tau _t (A^ * A) \geqq \tau _t (A)^ * \tau _t (A)$$ for allA∈? andt>0. Therefore one can define a contraction semigroupT on ?= \(\overline {\mathcal{M}\Omega } \) by $$T_t A\Omega = \tau _t (A)\Omega ,{\rm A} \in \mathcal{M},$$ where Ω is the cyclic and separating vector associated with ω. We prove 1. the fixed points ?(τ) of τ are given by ?(τ)=?∩T′=?∩E′, whereE is the orthogonal projection onto the subspace ofT-invariant vectors, 2. the state ω has a unique decomposition into τ-ergodic states if, and only if, ?(τ) or {?υE}′ is abelian or, equivalently, if (?, τ, ω) is ?-abelian, 3. the state ω is τ-ergodic if, and only if, ?υE is irreducible or if $$\mathop {\inf }\limits_{\omega '' \in Co\omega 'o\tau } \left\| {\omega '' - \omega '} \right\| = 0$$ for all normal states ω′ where Coω′°τ denotes the convex hull of {ω′°τ _t } _t>0. Subsequently we assume that τ is 2-positive,T is normal, andT* _t ?₊Ω \( \subseteqq \overline {\mathcal{M}_ + \Omega } \) , and then prove 4. there exists a strongly positive semigroup |τ| which commutes with τ and is determined by $$\left| \tau \right|_t \left( A \right)\Omega = \left| {T_t } \right|A\Omega ,$$ 5. results similar to 1 and 2 apply to |τ| but the τ-invariant state ω is |τ|-ergodic if, and only if, $$\mathop {\lim }\limits_{t \to \infty } \left\| {\omega 'o\tau _t - \omega } \right\| = 0$$ for all normal states ω′.     

Extension of quantum dynamical semigroups

We discuss the problem of extending a quantum dynamical semigroup to a group.     

One property of dynamical semigroups

We give a proof that the elements of a dynamical semigroup cannot generate s-reflections.     

Some remarks on dynamical semigroups

We deal with time evolution of a finite quantum system given by a dynamical semigroup Λ_t. For the semigroup we define and give some properties of the convex Λ_t-invariant subset of states “pathological” in some aspect evolving in strictly reversible manner independently of the stochastic surroundings of the system.     

Memory and Markov Blankets

In theoretical biology, we are often interested in random dynamical systems—like the brain—that appear to model their environments. This can be formalized by appealing to the existence of a (possibly non-equilibrium) steady state, whose density preserves a conditional independence between a biological entity and its surroundings. From this perspective, the conditioning set, or Markov blanket, induces a form of vicarious synchrony between creature and world—as if one were modelling the other. However, this results in an apparent paradox. If all conditional dependencies between a system and its surroundings depend upon the blanket, how do we account for the mnemonic capacity of living systems? It might appear that any shared dependence upon past blanket states violates the independence condition, as the variables on either side of the blanket now share information not available from the current blanket state. This paper aims to resolve this paradox, and to demonstrate that conditional independence does not preclude memory. Our argument rests upon drawing a distinction between the dependencies implied by a steady state density, and the density dynamics of the system conditioned upon its configuration at a previous time. The interesting question then becomes: What determines the length of time required for a stochastic system to ‘forget’ its initial conditions? We explore this question for an example system, whose steady state density possesses a Markov blanket, through simple numerical analyses. We conclude with a discussion of the relevance for memory in cognitive systems like us.     

Completely positive dynamical semigroups and quantum resonance theory

Starting from a microscopic system–environment model, we construct a quantum dynamical semigroup for the reduced evolution of the open system. The difference between the true system dynamics and its approximation by the semigroup has the following two properties: It is (linearly) small in the system–environment coupling constant for all times, and it vanishes exponentially quickly in the large time limit. Our approach is based on the quantum dynamical resonance theory.